Who can explain the incompatibilities between GR and QM for me?

[Holmes]

I'm somewhat familiar with GR but not all aspects of it, I do understand how the non-Euclidean nature of the 4D geometry enables us to consolidate gravitation and acceleration and the coordinate transformations, I studied this in some depth (and began with Eddington's space time and gravitation and the fascinating section on Gaussian curvature) but I'm not a mathematician and my knowledge is rather gappy (and nowadays rusty).

As for QM I read about that too but nowhere near as much effort was put in and this was all in the late 1970s when I had time to indulge in these.

So, having said all that - what is the simplest way to explain the deep incompatibilities between these two systems of thought?

[studiot]

43 minutes ago, Holmes said:

I'm somewhat familiar with GR but not all aspects of it, I do understand how the non-Euclidean nature of the 4D geometry enables us to consolidate gravitation and acceleration and the coordinate transformations, I studied this in some depth (and began with Eddington's space time and gravitation and the fascinating section on Gaussian curvature) but I'm not a mathematician and my knowledge is rather gappy (and nowadays rusty).

As for QM I read about that too but nowhere near as much effort was put in and this was all in the late 1970s when I had time to indulge in these.

So, having said all that - what is the simplest way to explain the deep incompatibilities between these two systems of thought?

First let me state quite clearly.

QM and GR are not necessarily incompatible.
Indeed they work quite well together in some cases.

But as you point out they are different systems of thought.

 

Now I digress a little to quote from your excellent thread in the science education section.

Quote

I work in technology and use the internet heavily so I'm aware of its nature,

I do not know what you mean by technology, but if you mean IT then you may find my offering easier to follow.

 

Eddington's book S, T & G is an excellent book and you will find nothing actually incorrect in it.

But it is almost one hundred years old now, and predated modern QM by a couple of decades.

Up to the 1930s, work on QM was based exclusively on extending classical non-relativistic mechanics to derive mathematically the observed phenomenon of quantisation.
Then in 1928 and through the 1930s Dirac introduced  relativistic wave equations to replace the schrodinger equation.
Developments have gone on ever since.

 

Now quantisation arises quite naturally in the solution of energy equations like schrodinger, which ignores gravitational forces as small compared to the electrostatic ones operating inside the atom.
But there are no (known) relativistic equations operating under gravity alone that result in quantisation in their solution.

So the big question is

Is gravity quantised, which under GR is effectively asking are space and time quantised or to put it another way are they granular?

Moving on a hundred years form Eddington we are still asking this question.

And an interesting modern book edited by Professor Shahn Majid explores where we are with this question.
If you understood S, T & G you will be able to follow this.

On Space and Time

Shahn Majid

Cambridge University Press  2008

 

Now asked if you were in IT since they have moved from the classic mathematics of continuity (analog computers) to discrete systems (digital computers)
Which is a parallel change.

 

The other big difference between QM and GR is the introduction of probability.

GR is a totally deterministic system of 'continuous' mathematics, using all the apparatus of topological continuity.

QM has a (highly successful)  interpretation in terms of probability theory. although it is often misapplied.

There are no probabilities in GR

 

Does this help ?

If you need clarification of anything (in particular I assume you understand when I say quantisation), please ask.

 

[Holmes]

34 minutes ago, studiot said:

First let me state quite clearly.

QM and GR are not necessarily incompatible.
Indeed they work quite well together in some cases.

But as you point out they are different systems of thought.

 

Now I digress a little to quote from your excellent thread in the science education section.

I do not know what you mean by technology, but if you mean IT then you may find my offering easier to follow.

 

Eddington's book S, T & G is an excellent book and you will find nothing actually incorrect in it.

But it is almost one hundred years old now, and predated modern QM by a couple of decades.

Up to the 1930s, work on QM was based exclusively on extending classical non-relativistic mechanics to derive mathematically the observed phenomenon of quantisation.
Then in 1928 and through the 1930s Dirac introduced  relativistic wave equations to replace the schrodinger equation.
Developments have gone on ever since.

 

Now quantisation arises quite naturally in the solution of energy equations like schrodinger, which ignores gravitational forces as small compared to the electrostatic ones operating inside the atom.
But there are no (known) relativistic equations operating under gravity alone that result in quantisation in their solution.

So the big question is

Is gravity quantised, which under GR is effectively asking are space and time quantised or to put it another way are they granular?

Moving on a hundred years form Eddington we are still asking this question.

And an interesting modern book edited by Professor Shahn Majid explores where we are with this question.
If you understood S, T & G you will be able to follow this.

On Space and Time

Shahn Majid

Cambridge University Press  2008

 

Now asked if you were in IT since they have moved from the classic mathematics of continuity (analog computers) to discrete systems (digital computers)
Which is a parallel change.

 

The other big difference between QM and GR is the introduction of probability.

GR is a totally deterministic system of 'continuous' mathematics, using all the apparatus of topological continuity.

QM has a (highly successful)  interpretation in terms of probability theory. although it is often misapplied.

There are no probabilities in GR

 

Does this help ?

If you need clarification of anything (in particular I assume you understand when I say quantisation), please ask.

 

Thank you for the reply.

Yes I understand that relativity is not a statistical theory but neither was Newtonian gravitation. On the other hand statistical mechanics (probabilistic models) predate quantum mechanics by at least fifty years, they were already around before the 1930s.

Because of this I don't understand why there's the view that GR and QM are fundamentally incompatible, we don't hear of statistical mechanics being incompatible with Newtonian mechanics for example.

I know there's a deep problem but I can't grasp it, can't see it - because of my very limited knowledge.

 

 

This paper seems to go there but won't be an easy read for me!

[MigL]

I know you are familiar with Richard Feynman, so you may be familiar with Feynman diagrams.
In Quantum Field Theoty all the connections in these diagrams represent possible interactions.
They are usually summed, then all the perturbative influences that give rise to infinities, are subtracted, by a process called re-normalization,to give the desired solution.

Gravity, on the other hand, is self interacting ( that means gravity gravitates ), and the resultant infinities from the perturbative summng gets out of control, such that they cannot be removed by re-normalization.

Any time infinities result in a solution, you have exceeded the bounds of applicability of your theory.
So, either perturbative quantum field theory or re-normalization are not applicable to gravity.
Therein lies the incompatibility.

Even more simply ...
In QFT, space-time is simply the stage on which events happen, and the events have no affect on the stage.
In GR, space-time is the field that governs how events unfold, but it is affected, and a participant, in those events.

The Wiki entry on Quantum Gravity may be helpful to you

Quantum gravity - Wikipedia

Edited June 30, 2021 by MigL

[studiot]

17 minutes ago, Holmes said:

Thank you for the reply.

I understand, thanks but no thanks.

[Holmes]

15 minutes ago, MigL said:

I know you are familiar with Richard Feynman, so you may be familiar with Feynman diagrams.
In Quantum Field Theoty all the connections in these diagrams represent possible interactions.
They are usually summed, then all the perturbative influences that give rise to infinities, are subtracted, by a process called re-normalization,to give the desired solution.

Gravity, on the other hand, is self interacting ( that means gravity gravitates ), and the resultant infinities from the perturbative summng gets out of control, such that they cannot be removed by re-normalization.

Any time infinities result in a solution, you have exceeded the bounds of applicability of your theory.
So, either perturbative quantum field theory or re-normalization are not applicable to gravity.
Therein lies the incompatibility.

Even more simply ...
In QFT, space-time is simply the stage on which events happen, and the events have no affect on the stage.
In GR, space-time is the field that governs how events unfold, but it is affected, and a participant, in those events.

Thanks, yes I've seen Feynman Diagrams (and often wondered what all the fuss was about them - but then read there was something quite profound going on) and read about renormalization (my mathematical knowledge is a serious limitation here).

But I was just reading this after reading your post:

and that is very very interesting, I've not really seen this problem or read much about it before.

[MigL]

I have not done these calculations myself, but the electromagnetic mass, which includes the mass-energy of its electrostatic field, and is usually thought of as a cloud of virtual particles surrounding the charged particle, is exactly what gets modified by re-normalization, such that the actual mass is left.
This is not the same self- interaction that gravity xperiences, and, as a result, re-normalization fails to remove the infinities.
The difference between the self interactions is that an EM field has mass-energy, but doesn't produce more charge, while a gravitational field has mass-energy, which produces more gravity, which produces more field mass-energy, which produces more gravity, which ....
Do you see what I mean ?

[Holmes]

Just now, MigL said:

I have not done these calculations myself, but the electromagnetic mass, which includes the mass-energy of its electrostatic field, and is usually thought of as a cloud of virtual particles surrounding the charged particle, is exactly what gets modified by re-normalization, such that the actual mass is left.
This is not the same self- interaction that gravity xperiences, and, as a result, re-normalization fails to remove the infinities.
The difference between the self interactions is that an EM field has mass-energy, but doesn't produce more charge, while a gravitational field has mass-energy, which produces more gravity, which produces more field mass-energy, which produces more gravity, which ....
Do you see what I mean ?

Kinda...!

I'm reading about electromagnetic mass, not really read about this before. It begs the question (in my naive mind) why regard the charged particle as having a mass + an electromagnetic mass? why not regard all of its mass as electromagnetic mass, rather than a particle + a field why not just think of it as a field with mass...

This is naive I know, but it is quite interesting and is giving a bit more insight so thanks.

 

 

[MigL]

1 minute ago, Holmes said:

why not just think of it as a field with mass...

QFT does.
Quantum particles are simply excitations of the field, above a threshold of action.
Those excitations below the threshold, are virtual particles.

[md65536]

1 hour ago, Holmes said:

Classically, a point particle with a precisely located energy makes sense. With QM wouldn't such a thing require a completely undefined momentum, maybe other consequences? Wouldn't you need to start with a particle with infinite energy in QM as well? This example sounds like less of an incompatibility of the two, and more a problem of applying certain aspects of modern physics while ignoring others, and as expected reaching a conclusion that doesn't agree with measurements of reality.

[beecee]

3 hours ago, MigL said:

Gravity, on the other hand, is self interacting ( that means gravity gravitates

This is what is known as "nonlinearity" correct? or gravity makes gravity.

So in essence the incompatibilty is one treats gravity as discrete, the other continious, correct? 

So what about concepts of Planck limits? My thinking is that these are simply useful manmade concepts. So what will a validated QGT entail do you think?

Good question by the way and good answers thus far..

[mathematic]

A major problem between them is trying to describe what goes on inside a black hole.  GR ends up as a singularity. QM says impossible.

[MigL]

5 hours ago, MigL said:

The Wiki entry on Quantum Gravity may be helpful to you

Quantum gravity - Wikipedia

 

[Kartazion]

38 minutes ago, MigL said:

The Wiki entry on Quantum Gravity may be helpful to you

Quantum gravity - Wikipedia

I don't understand everything, but I think that there is a reservation in relation to the singularity for the Loop Quantum Gravity theory.

On the other hand and as a singularity, we are talking about singularity avoidance in Loop Quantum Cosmology. I do not know why.
 

[beecee]

3 hours ago, mathematic said:

A major problem between them is trying to describe what goes on inside a black hole.  GR ends up as a singularity. QM says impossible.

GR tells us that once the Schwarzchild radiu/limit is reached, that further collapse is compulsory, but at the same time breaks down at the quantum/Planck level, I believe is correct. So a QGT takes over eliminating infinities. Most physicists today dismiss the Singularity of infinte density and curvature anyway. 

Awesome stuff anyway of what I am able to comprehend!!

[joigus]

I think the question can be approached on a number of levels. For starters, quantum mechanics makes time a very special parameter. You need a distinguished time that goes hand in hand with a so-called Hamiltonian of the system (the energy operator). This Hamiltonian is also the mathematical operation that embodies time translation for the system.

GR, on the contrary, has no special time. There is no preferred coordinate system in GR. If you have no special time, you have no special Hamiltonian, which means you have no special time-updating law for the state. So right from the start, the symmetries of GR don't bode well with the special needs of QM.

This difficulty though, can be overcome AFAIK. One formalism that does it is the so-called Ashtekar variables, that allow you to include all the constraints of GR into a significant set of variables that are amenable to quantization. And they give you a Hamiltonian. But a lot of preliminary work is needed to get there. In any case, as @studiot suggests, the conceptual bases are quite different.

Then there is the question of the perturbative structure of the theory, as @MigL pointed out. The theory is non-linear, has more field variables, and the loop calculations become intractable pretty soon. This, in and of itself, would not be catastrophic, as Yang-Mills fields (in the non-Abelian case, which is strong nuclear force) are also self-interacting and have a richer field-variable structure. But YM fields are far better-behaved than gravitation at short-distance (large-momenta) scales. The theory is free at short distances (large momenta), while gravity is just the opposite. Gravity, also, has no polarity and is thermodynamically exceptional.

This bad large-momentum behaviour connects with what today is considered the ultimate reason why gravity is so unwieldy to a quantum treatment: the dimensions of the gravitational coupling constant. It is dimensionful (and badly so), as opposed to the dimensionless character of QED, QCD, and EW coupling constants.

Renormalization crudely consists in decreeing a maximum momentum Λ  for every scale that we wish to study, and then prove that the observables inferred from the quantum scattering amplitudes can be expanded as a sum of two parts, one that remains under control (finite), plus a logarithmically divergent one (scale independent). Now, you cannot do that with gravity. There are two technical ways to characterize this in words:

1) Gravity doesn't look like a scale-independent quantum field theory at large momenta

2) The large-energy spectrum of gravity is black-hole dominated

Equivalence of both is discussed at: https://arxiv.org/pdf/0709.3555.pdf

It's quite technical, in spite of the title, but enough words can be found there so that a crude idea of what goes on can be obtained.

There is a last-ditch attempt in QFT to make quantum gravity renormalizable, and that's called asymptotic safety, initiated by Steven Weinberg, and it's based on hoping for a point in the phase space where this bad behaviour is saved by a so-called fixed point of the beta function (a function that monitors the renormalization behaviour when you shift the cutoff). But it takes a lot of guesswork and I don't know what the state of the art is at this point.

Then comes supersymmetry. It's a big hope, because supersymmetry leads to much cancellation of infinities.

New technologies have been developed in recent years, like calculations using the formalism of maximally-helicity-violating scattering amplitudes. It's a technology that involves massless gauge bosons, always leads to finite calculations, but is considerably more abstract, because it uses expansions of the amplitudes that cannot be understood as local quantities (at a point), and it involves twistors, which entails expanding space-time points into pairs of massless spin-1/2 wavefunctions (Weyl spinors).

Two interesting lectures on the subject. The 1st one is more technical, but again, enough worded arguments are given so that one can get an idea of what goes on.

Quantum gravity and its discontents (by Stanley Deser, 2010): 

Quantizing gravity and why it is difficult (Leonard Susskind, 2013):

 

[Holmes]

56 minutes ago, joigus said:

I think the question can be approached on a number of levels. For starters, quantum mechanics makes time a very special parameter. You need a distinguished time that goes hand in hand with a so-called Hamiltonian of the system (the energy operator). This Hamiltonian is also the mathematical operation that embodies time translation for the system.

GR, on the contrary, has no special time. There is no preferred coordinate system in GR. If you have no special time, you have no special Hamiltonian, which means you have no special time-updating law for the state. So right from the start, the symmetries of GR don't bode well with the special needs of QM.

This difficulty though, can be overcome AFAIK. One formalism that does it is the so-called Ashtekar variables, that allow you to include all the constraints of GR into a significant set of variables that are amenable to quantization. And they give you a Hamiltonian. But a lot of preliminary work is needed to get there. In any case, as @studiot suggests, the conceptual bases are quite different.

Then there is the question of the perturbative structure of the theory, as @MigL pointed out. The theory is non-linear, has more field variables, and the loop calculations become intractable pretty soon. This, in and of itself, would not be catastrophic, as Yang-Mills fields (in the non-Abelian case, which is strong nuclear force) are also self-interacting and have a richer field-variable structure. But YM fields are far better-behaved than gravitation at short-distance (large-momenta) scales. The theory is free at short distances (large momenta), while gravity is just the opposite. Gravity, also, has no polarity and is thermodynamically exceptional.

This bad large-momentum behaviour connects with what today is considered the ultimate reason why gravity is so unwieldy to a quantum treatment: the dimensions of the gravitational coupling constant. It is dimensionful (and badly so), as opposed to the dimensionless character of QED, QCD, and EW coupling constants.

Renormalization crudely consists in decreeing a maximum momentum Λ  for every scale that we wish to study, and then prove that the observables inferred from the quantum scattering amplitudes can be expanded as a sum of two parts, one that remains under control (finite), plus a logarithmically divergent one (scale independent). Now, you cannot do that with gravity. There are two technical ways to characterize this in words:

1) Gravity doesn't look like a scale-independent quantum field theory at large momenta

2) The large-energy spectrum of gravity is black-hole dominated

Equivalence of both is discussed at: https://arxiv.org/pdf/0709.3555.pdf

It's quite technical, in spite of the title, but enough words can be found there so that a crude idea of what goes on can be obtained.

There is a last-ditch attempt in QFT to make quantum gravity renormalizable, and that's called asymptotic safety, initiated by Steven Weinberg, and it's based on hoping for a point in the phase space where this bad behaviour is saved by a so-called fixed point of the beta function (a function that monitors the renormalization behaviour when you shift the cutoff). But it takes a lot of guesswork and I don't know what the state of the art is at this point.

Then comes supersymmetry. It's a big hope, because supersymmetry leads to much cancellation of infinities.

New technologies have been developed in recent years, like calculations using the formalism of maximally-helicity-violating scattering amplitudes. It's a technology that involves massless gauge bosons, always leads to finite calculations, but is considerably more abstract, because it uses expansions of the amplitudes that cannot be understood as local quantities (at a point), and it involves twistors, which entails expanding space-time points into pairs of massless spin-1/2 wavefunctions (Weyl spinors).

Two interesting lectures on the subject. The 1st one is more technical, but again, enough worded arguments are given so that one can get an idea of what goes on.

Quantum gravity and its discontents (by Stanley Deser, 2010): 

Quantizing gravity and why it is difficult (Leonard Susskind, 2013):

 

Very detailed, thanks for the lectures too, I doubt I'll follow very far but I'm sure I'd pick up some insights here and there!

[joigus]

16 hours ago, Holmes said:

Very detailed, thanks for the lectures too, I doubt I'll follow very far but I'm sure I'd pick up some insights here and there!

I recommend you Susskind's Theoretical Minimum (Quantum Mechanics). You should have no problem following the main ideas and equations. Then his lectures on QFT to get an idea of what this renormalisation business is all about.

I picture you as the ideal person for his approach: with a sound knowledge of physics but having fallen somewhat out of touch.

The other lectures I provided are just to get a flavour of what the problems are. They're difficult for me too.

I think the right way to tackle theoretical physics lectures is: I know I won't digest 100 % of this material. Let's see if I can understand the key ideas. And keep going. Things start clicking after a while. You probably know very well from your experience studying GR years ago.

You can always forget about the equations for a while and concentrate on the words.